An exact Kerr-like solution has been obtained recently in Einstein-bumblebee gravity model where Lorentz symmetry is spontaneously broken. In this paper, we investigate the superradiant instability of the Kerr-like black hole under the perturbation of a massive scalar field. We find the Lorentz breaking parameter $L$ does not affect the superradiance regime or the regime of the bound states. However, since $L$ appears in the metric and its effect cannot be erased by redefining the rotation parameter $\tilde{a}=\sqrt{1+L}a$, it indeed affects the bound state spectrum and the superradiance. We calculate the bound state spectrum via the continued-fraction method and show the influence of $L$ on the maximum binding energy and the damping rate. The superradiant instability could occur since the superradiance condition and the bound state condition could be both satisfied. Compared with Kerr black hole, the nature of the superradiant instability of this black hole depends nonmonotonously not only on the rotation parameter of the black hole $\tilde{a}$ and the product of the black hole mass $M$ and the field mass $\mu$, but also on the Lorentz breaking parameter $L$. Through the Monte Carlo method, we find that for the $l=m=1$ state the most unstable mode occurs at $L=-0.79637$, $\tilde{a}/M=0.99884$, and $M\mu =0.43920$, with the maximum growth rate of the field ${\omega }_{I}M=1.676\times {10}^{-6}$, which is about 10 times than that in the Kerr black hole.

中文翻译：

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爱因斯坦大黄蜂引力中类克尔黑洞的超辐射不稳定性

最近在爱因斯坦-大黄蜂引力模型中获得了一个精确的类克尔解，其中洛伦兹对称性自发破缺。在本文中，我们研究了类克尔黑洞在巨大标量场扰动下的超辐射不稳定性。我们找到洛伦兹破坏参数$这$不影响超辐射机制或束缚态机制。然而，由于$这$ 出现在度量中，并且无法通过重新定义旋转参数来消除其影响 $\widetilde{\mathrm{一种}}=\sqrt{1+这}\mathrm{一种}$，它确实影响了束缚态谱和超辐射。我们通过连续分数法计算束缚态谱，并展示了$这$最大结合能和阻尼率。由于超辐射条件和束缚态条件都可以满足，因此可能发生超辐射不稳定。与克尔黑洞相比，该黑洞超辐射不稳定性的性质非单调地不仅取决于黑洞的旋转参数$\widetilde{\mathrm{一种}}$ 和黑洞质量的乘积 $米$ 和场质量 $\mu$，而且还有洛伦兹破断参数 $这$. 通过蒙特卡罗方法，我们发现对于$这=米=1$ 状态最不稳定的模式发生在 $这=-0.79637$, $\widetilde{\mathrm{一种}}/米=0.99884$， 和 $米\mu =0.43920$, 具有最大增长率的领域 ${\omega }_{\mathrm{一世}}米=1.676\times {10}^{-6}$，大约是克尔黑洞的 10 倍。